Overview

The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. It is shown that these algebras have natural integral forms which can be specialised at roots of 1 and yield new objects which include quantum versions of the semisimple groups over fields of positive characteristic. The theory of quantum group has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical bases with rather remarkable properties. The book contains an extensive treatment of the theory of canonical bases, in the framework of perverse sheaves. The theory developed in the book includes the case of quantum affine enveloping algebras and, more generally, the quantum analogs of the Kac-Moody Lie algebras. This book may be of interest to mathematicians working in representation theory of Lie groups and Lie algebras, knot theorists, and to. theoretical physicists. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used by graduate students, or as a textbook.

Full Product Details

Author:   George Lusztig
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Edition:   1st ed. 1993
Volume:   110
Weight:   0.675kg
ISBN:  

9780817637125

ISBN 10:   0817637125
Pages:   366
Publication Date:   01 April 1994
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Out of Stock Indefinitely
Availability:   Out of stock  

Table of Contents

THE DRINFELD JIMBO ALGERBRA U.- The Algebra f.- Weyl Group, Root Datum.- The Algebra U.- The Quasi--Matrix.- The Symmetries of an Integrable U-Module.- Complete Reducibility Theorems.- Higher Order Quantum Serre Relations.- GEOMETRIC REALIZATION OF F.- Review of the Theory of Perverse Sheaves.- Quivers and Perverse Sheaves.- Fourier-Deligne Transform.- Periodic Functors.- Quivers with Automorphisms.- The Algebras and k.- The Signed Basis of f.- KASHIWARAS OPERATIONS AND APPLICATIONS.- The Algebra .- Kashiwara's Operators in Rank 1.- Applications.- Study of the Operators .- Inner Product on .- Bases at ?.- Cartan Data of Finite Type.- Positivity of the Action of Fi, Ei in the Simply-Laced Case.- CANONICAL BASIS OF U.- The Algebra .- Canonical Bases in Certain Tensor Products.- The Canonical Basis .- Inner Product on .- Based Modules.- Bases for Coinvariants and Cyclic Permutations.- A Refinement of the Peter-Weyl Theorem.- The Canonical Topological Basis of .- CHANGE OF RINGS.- The Algebra .- Commutativity Isomorphism.- Relation with Kac-Moody Lie Algebras.- Gaussian Binomial Coefficients at Roots of 1.- The Quantum Frobenius Homomorphism.- The Algebras .- BRAID GROUP ACTION.- The Symmetries of U.- Symmetries and Inner Product on f.- Braid Group Relations.- Symmetries and U+.- Integrality Properties of the Symmetries.- The ADE Case.

Reviews

From the reviews: There is no doubt that this volume is a very remarkable piece of work...Its appearance represents a landmark in the mathematical literature. -Bulletin of the London Mathematical Society This book is an important contribution to the field and can be recommended especially to mathematicians working in the field. -EMS Newsletter The present book gives a very efficient presentation of an important part of quantum group theory. It is a valuable contribution to the literature. -Mededelingen van het Wiskundig Lusztig's book is very well written and seems to be flawless...Obviously, this will be the standard reference book for the material presented and anyone interested in the Drinfeld-Jimbo algebras will have to study it very carefully. -ZAA [T]his book is much more than an 'introduction to quantum groups.' It contains a wealth of material. In addition to the many important results (of which several are new-at least in the generality presented here), there are plenty of useful calculations (commutator formulas, generalized quantum Serre relations, etc.). -Zentralblatt MATH George Lusztig lays out the large scale structure of the discussion that follows in the 348 pages of his Introduction to Quantum Groups. ... A significant and important work. ... it's terrific stuff, elegant and deep, and Lusztig presents it very well indeed, of course. (Michael Berg, The Mathematical Association of America, January, 2011)

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